In this sense, Eq. (11.60), and hence curve C in Fig. 11.6, is a type of “universal relation” which can be used for all airfoils. Equation (11.60), in conjunction with anyone of the compressibility corrections given by Eqs. (11.51), (11.54), or (11.55), allows us to estimate the critical Mach number for a given airfoil as follows:

1. By some means, either experimental or theoretical, obtain the low-speed incompressible value of the pressure coefficient Cp,o at the minimum pressure point on the given airfoil.

2. Using any of the compressibility corrections, Eq. (11.51), (11.54), or (11.55), plot the variation of Cp with M¥. This is represented by curve B in Fig. 11.6.

3. Somewhere on curve B, there will be a single point where the pressure coefficient corresponds to locally sonic flow. Indeed, this point must coincide with Eq. (11.60), represented by curve C in Fig. 11.6. Hence, the intersection of curves Band C represents the point corresponding to sonic flow at the minimum pressure location on the airfoil. In turn, the value of M¥ at this intersection is, by definition, the critical Mach number, as shown in Fig. 11.6.

The graphical construction in Fig. 11.6 is not an exact determination of Mcr.

Although curve C is exact, curve B is approximate because it represents the approximate compressibility correction. Hence, Fig. 11.6 gives only an estimation of Mcr. However, such an estimation is quite useful for preliminary design, and the results from Fig. 11.6 are accurate enough for most applications.

Consider two airfoils, one thin and the other thick, as sketched in Fig. 11.7. First consider the low-speed incompressible flow over these airfoils. The flow over the thin airfoil is only slightly perturbed from the freestream. Hence, the expansion over the top surface is mild, and Cp,o at the minimum pressure point is a negative number of only small absolute magnitude, as shown in Fig. 11.7.

DRAG-DIVERGENCE MACH NUMBER:

THE SOUND BARRIER

THE AREA RULE

For a moment, let us expand our discussion from two-dimensional airfoils to a consideration of a complete airplane. In this section, we introduce a design concept which has effectively reduced the drag rise near Mach 1 for a complete airplane.

As stated before, the first practical jet-powered aircraft appeared at the end of World War II in the form of the German Me 262. This was a subsonic fighter plane with a top speed near 550 mi/h. The next decade saw the design and production of many types of jet aircraft-all limited to subsonic flight by the large drag near Mach 1. Even the “century” series of fighter aircraft designed to give the U.S. Air Force supersonic capability in the early 1950s, such as the Convair F-1 02 delta-wing airplane, ran into difficulty and could not at first readily penetrate the sound barrier in level flight. The thrust of jet engines at that time simply could not overcome the large peak drag near Mach 1.

A planview, cross section, and area distribution (cross-sectional area versus distance along the axis of the airplane) for a typical airplane of that decade are sketched in Fig. 11.10. Let A denote the total cross-sectional area at any given station. Note that the cross-sectional area distribution experiences some abrupt changes along the axis, with discontinuities in both A and dA/ dx in the regions of the wing.

In contrast, for almost a century, it was well known by ballisticians that the speed of a supersonic bullet or artillery shell with a smooth variation of cross-sectional area was higher than projectiles with abrupt or discontinuous area

distributions. In the mid-1950s, an aeronautical engineer at the NACA Langley Aeronautical Laboratory, Richard T.

Whitcomb, put this knowledge to work on the problem of transonic flight of airplanes. Whitcomb reasoned that the variation of cross-sectional area for an airplane should be smooth, with no discontinuities.

This meant that, in the region of the wings and tail, the fuselage cross-sectional area should decrease to compensate for the addition of the wing and tail cross sectional area. This led to a “coke bottle” fuselage shape, as shown in Fig. 11.11. Here, the plan view and area distribution are shown for an aircraft with a relatively smooth variation of A(x). This design philosophy is called the area rule, and it successfully reduced the peak drag near Mach 1 such that practical airplanes could fly supersonically by the mid-1950s. The variations of drag coefficient with M¥ for an area-ruled and non-area-ruled airplane are schematically compared in Fig. 11.12; typically, the area rule leads to a factor-of-2 reduction in the peak drag near Mach 1.

The development of the area rule was a dramatic breakthrough in high-speed flight, and it earned a substantial reputation for Richard Whitcomb-a reputation which was to be later garnished by a similar breakthrough in transonic airfoil design, to be discussed in Sec. 11.9. The original work on the area rule was presented by Whitcomb in Ref. 31, which should be consulted for more details.